General Equilibrium - Illustration (Max and Min Functions)

By Aleesha Mary Joseph

FINDING OPTIMAL DEMAND FOR GOOD X AND GOOD Y WHEN THE UTILITY FUNCTION OF THE CONSUMER IS
                                                
  U = min{ X+2Y , 2X +Y }  

Subject to the budget constraint  Px X  +  Py Y  =  M  
where Px is the price of good X and Py the price of good Y.

Step 1

Let us first figure out the indifference curve(IC) pertaining to this particular utility function. We know that IC is the combination of all those bundles (x,y) which gives same utility to the consumer. So let us keep utility constant at say 10. Then now our task is to plot  min{ X+2Y , 2X +Y }  = 10 on the X-Y space. If min{ X+2Y , 2X +Y }  = 10, then either (i) X+2Y  = 10  OR  (ii) 2X+Y  = 10.  Let us plot (i) & (ii) on the same graph:

Note that (i) & (ii) intersects when X=Y. And slope of (i) is ½ and slope of (ii) is 2.

 Now note that the yellow region corresponds to X + 2Y < 10. Also at any point on the red line 2X + Y =10. These statements imply that at any point on the red line min{ X+2Y , 2X +Y }  = min{ <10 , 10 }. ie min{ X+2Y , 2X +Y } < 10. This implies that the red line is not part of the IC representing 10 utils. 

 Similarly the orange region corresponds to 2X + Y < 10. Also at any point on the blue line X + 2Y =10. These statements imply that at any point on the red line min{ X+2Y , 2X +Y }  = min{ 10 , <10 }. ie min{ X+2Y , 2X +Y } < 10. This implies that the blue line is not part of the IC representing 10 utils. 

Note that the grey region corresponds to X + 2Y > 10. Also at any point on the purple line 2X + Y =10. These statements imply that at any point on the purple line min{ X+2Y , 2X +Y }  = min{ >10 , 10 }. ie min{ X+2Y , 2X +Y } = 10. This implies that the purple line is part of the IC representing 10 utils. 

Similarly, the blue region corresponds to 2X + 2Y > 10. Also at any point on the orange line X + 2Y =10. These statements imply that at any point on the orange line min{ X+2Y , 2X +Y }  = min{ 10 , >10 }. ie min{ X+2Y , 2X +Y } = 10. This implies that the orange line is part of the IC representing 10 utils

Thus from the above arguments it is clear that the points on the red and blue lines are not part of the IC that gives 10 utils. That is those points do not solve min{ X+2Y , 2X +Y }  = 10. Therefore let us erase the red and blue line segments from our figure. While the points on the purple and orange lines solve min{ X+2Y , 2X +Y }  = 10.

 The figure ABC represents all those bundles (x,y) which satisfy min{ X+2Y , 2X +Y }  = 10. Hence ABC is the IC which represents 10 utils. Note that slope of BC is ½ and slope of AB is 2.
The Indifference map is as follows:

Now let us consider various price ratios and figure out where the consumer will consume.


Step 2

(1) If Px/Py < ½ then budget line would look like the red line in the following figure:

The highest IC that touches the budget line is the blue colored one and hence the optimal consumption will be at E. At E consumer spends his entire income on good X and consumes zero of the other good.

(2) If Px/Py >2 then budget line would look like the red line in the following figure. The highest IC that touches the budget line is the blue colored one and hence the optimal consumption will be at F. At F consumer spends his entire income on good Y and consumes zero of good X

                                     

(3) If ½<Px/Py <2 then budget line would look like the red line in the Figure 10 The highest IC that touches the budget line is the blue colored one and hence the optimal consumption will be at G. We know that at the kink G X=Y. Substitute this in the budget constraint. Then we will get the optimal consumption as X = Y = M/(Px + Py).

                                      

(4) If Px/Py = ½, then any point on BC is optimal. And any point on BC will satisfy X + 2Y = 10 & the inequality X≥Y.

                       

(5) Similarly when Px/Py = 2 any point on AB is optimal. And any point on AB will satisfy 2X +Y = 10 & the inequality Y≥X.


To summarize we can write demand for x & y as follows:
(x,y)  =   (M/Px, 0)                                                                          when Px/Py  < ½
              All (x,y) such that X + 2Y = 10 & X≥Y                                  when Px/Py  =½
             (M/(Px+Py), M/(Px+Py))                                                    when ½<Px/Py<2
              All (x,y) such that 2X + Y = 10 & Y≥X                                  when Px/Py  =2
              (0 , M/Py)                                                                        when Px/Py  >2



FINDING OPTIMAL DEMAND FOR GOOD X AND GOOD Y WHEN THE UTILITY FUNCTION OF THE CONSUMER IS
                                                  U = max{ X+2Y , 2X +Y }  


Using the same technique of analysis as in the previous question, the IC map of the max function would be as follows:

Exercise: Consider ABC is the IC representing 10 utils. Then Find the coordinates of the points A & C. Now verify that the slope of AC is 1. Convince yourself that for any given IC the slope of the line segment joining the end points of the IC (like A & C) is 1.

1) When Px/Py<1ie slope of the budget line (red line) is less than 1.

Clearly consumer will optimally choose the bundle C.



2) When Px/Py>1ie slope of the budget line (red line) is greater than 1.

In this case optimal consumption will be at A.


3) When Px/Py=1ie slope of the budget line (red line) is equal to 1.

In this case optimal consumption will be at A & C.


To summarize we can write demand for x & y as follows:
(x,y)  =   (M/Px, 0)                                                                        when Px/Py  < 1
               (M/Px, 0) or (0, M/ Py )                                                  when Px/Py  =1
               (0, M/ Py )                                                                     when Px/Py  >1


(Aleesha Mary Joseph graduated from St. Stephen's College in 2013. She is currently pursuing MA in Economics at Delhi School of Economics)